Linear Algebraic Groups - Oklahoma State University–Stillwatermath.okstate.edu/people/pyan/LinearAlgebraicGroups.pdf · 2015-10-28 · Linear Algebraic Groups Fall 2015 These are

Linear Algebraic Groups

Fall 2015

These are notes for the graduate course Math 6690 (Linear Algebraic Groups) taughtby Dr. Mahdi Asgari at the Oklahoma State University in Fall 2015. The notes are takenby Pan Yan ([email protected]), who is responsible for any mistakes. If you noticeany mistakes or have any comments, please let me know.

Contents

1 Root Systems (08/19) 2

2 Review of Algebraic Geometry I (08/26) 13

3 Review of Algebraic Geometry II, Introduction to Linear AlgebraicGroups I (09/02) 17

4 Introduction to Linear Algebraic Groups II (09/09) 23

5 Introduction to Linear Algebraic Groups III (09/16) 29

6 Jordan Decomposition (09/23) 33

7 Commutative Linear Algebraic Groups (09/30) 38

8 Tori and Torus Actions (10/07) 44

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1 Root Systems (08/19)

Root Systems

Reference for this part is Lie Groups and Lie Algebras, Chapters 4-6 by N. Bourbaki.Let V be a finite dimensional vector space over R. An endomorphism s : V → V is

called a reflection if there exists 0 6= a ∈ V such that s(a) = −a and s fixes pointwise ahyperplane (i.e., a subspace of codimension 1) in V . Then

V = ker(s− 1)⊕ ker(s+ 1)

and s2 = 1. We denote V +s = ker(s− 1) which is a hyperplane in V , and V −s = ker(s+ 1)

which is just Ra.Let D = im(1− s), then dim(D) = 1. This implies that given 0 6= a ∈ D, there exists

a nonzero linear form a∗ : V → R such that

x− s(x) = 〈x, a∗〉 a,∀x ∈ V

where 〈x, a∗〉 = a∗(x). Conversely, given some 0 6= a ∈ V and a linear form a∗ 6= 0 on V ,set

sa,a∗(x) = x− 〈x, a∗〉 a,∀x ∈ V

this gives an endomorphism of V such that 1− sa,a∗ is of rank 1. Note that

s2a,a∗(x) = sa,a∗(x− 〈x, a∗〉 a)

= x− 〈x, a∗〉 a− 〈x− 〈x, a∗〉 a, a∗〉 a= x− 2 〈x, a∗〉 a+ 〈x, a∗〉 〈a, a∗〉 a= x+ (〈a, a∗〉 a− 2) 〈x, a∗〉 a.

So sa,a∗ is a reflection if and only if 〈a, a∗〉 = 2, i.e., sa,a∗(a) = −a.WARNING: 〈x, a∗〉 is only linear in the first variable, but not the second.

Remark 1.1. (i) When V is equipped with a scalar product (i.e., a non-degenerate sym-metric bilinear form B), then we can consider the so called orthogonal reflections, i.e., thefollowing equivalent conditions hold:

V +s and V −s are perpendicular w.r.t. B ⇔ B is invariant under s.

In that case,

s(x) = x− 2B(x, a)

B(a, a)a.

(ii) A reflection s determines the hyperplane uniquely, but not the choice of the nonzeroa (but it does in a root system, which we will talk about later).

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Definition 1.2. Let V be a finite dimensional vector space over R, and let R be a subsetof V . Then R is called a root system in V if(i) R is finite, 0 6∈ R, and R spans V ;(ii) For any α ∈ R, there is an α∨ ∈ V ∗ where V ∗ = {f : V → R linear} is the dual of V ;(iii) For any α ∈ R, α∨(R) ⊂ Z.

Lemma 1.3. Let V be a vector space over R and let R be a finite subset of V generatingV . For any α ∈ R such that α 6= 0, there exists at most one reflection s of V such thats(α) = −α and s(R) = R.

Proof. Suppose there are two reflections s, s′ such that s(α) = s′(α) = −α and s(R) =s′(R) = R. Then s(x) = x−f(x)α, s′(x) = x− g(x)α for some linear functions f(x), g(x).Since s(α) = s′(α) = −α, we have f(α) = g(α) = 2. Then

s(s′(x)) = x− g(x)α− f (x− g(x)α)α

= x− g(x)α− f(x)α+ f(α)g(x)α

= x− g(x)α− f(x)α+ 2g(x)α

= x− (g(x)− f(x))α

is a linear function, and s(s′(R)) = R. Since R is finite, s ◦ s′ is of finite order, i.e.,(s ◦ s′)n = (s ◦ s′) ◦ (s ◦ s′) ◦ · · · ◦ (s ◦ s′) is identity for some n ≥ 1. Moreover,

(s ◦ s′)2(x) = x− (g(x)− f(x))α− (g(x− (g(x)− f(x))α)− f(x− (g(x)− f(x))α))α

= x− 2(g(x)− f(x))α

and by applying the composition repeatedly, we have

(s ◦ s′)n(x) = x− n(g(x)− f(x))α.

But (s ◦ s′)n(x) = x for all x ∈ V , therefore, g(x) = f(x). Hence s(x) = s′(x).

Lemma 1.3 shows that given α ∈ R, there is a unique reflection s of V such thats(α) = −α and s(R) = R. That implies α determines sα,α∨ and α∨ uniquely, and hence(iii) in the definition makes sense.

We can write sα,α∨ = sα. Then

sα(x) = x−⟨x, α∨

⟩α,∀x ∈ V.

The elements of R are called roots (of this system). The rank of the root system is thedimension of V . We define

A(R) = finite group of automorphisms of V leaving R stable

and the Weyl group of the root system R to be

W = W (R) = the subgroup of A(R) generated by the sα, α ∈ R.

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Remark 1.4. Let R be a root system in V . Let (x|y) be a symmetric bilinear form on V ,non-degenerate and invariant under W (R). We can use this form to identify V with V ∗.Now if α ∈ R, then α is non-isotropic (i.e., (α|α) 6= 0) and

α∨ =2α

(α|α).

This is because we saw that (x|y) invariant under sα implies

sα(x) = x− 2(x|α)

(α|α)α.

Proposition 1.5. R∨ = {α∨ : α ∈ R} is a root system in V ∗ and α∨∨ = α, ∀α ∈ R.

Proof. (Sketch). For (i) in Definition 1.2, R∨ is finite and does not contain 0. To see thatR∨ spans V ∗, we need to use the canonical bilinear form on V × V ∗ to identify

VQ = Q− vector space of V generated by the α

andV ∗Q = Q− vector space of V ∗ generated by the α∨

with the dual of the other. This way, the α∨ generate V ∗.For (ii) in Definition 1.2, sα,α∨ is an automorphism of V equipped with the root system

R and t(sα,α∨)−1 leaved R∨ stable, but one can check that t(sα,α∨)−1 = sα,α∨ and α∨∨ = α.For (iii) in Definition 1.2, note that 〈β, α∨〉 ∈ Z ∀β ∈ R,∀α∨ ∈ R∨, so R∨ satisfies

(iii).

Remark 1.6. R∨ is called the dual root system of R. The map α 7→ α∨ is a bijectionfrom R to R∨ and is called the canonical bijection from R to R∨.

WARNING: If α, β ∈ R and α+ β ∈ R, then (α+ β)∨ 6= α∨ + β∨ in general.

Remark 1.7. (i) The facts sα(α) = −α and sα(R) ⊂ R imply R = −R.(ii) It is also clear that (−α)∨ = −α∨. −1 ∈ A(R), but -1 is not always an element of

W (R).(iii) The equality t(sα,α∨)−1 = sα∨,α implies the map u 7→ tu−1 is an isomorphism from

W (R) to W (R∨), so we can identify these two via this isomorphism, and simply considerW (R) as acting on both V and V ∗. It is similar for A(R).

First Examples

Now we give a few examples of root systems.

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Example 1.8. (A1): V = Re. The root system is

R = {α = e,−e}.

The reflection is sα(x) = −x. V +s = 0, V −s = V . A(R) = W (R) = Z/2Z. The usual scalar

product (x|y) = xy is W (R)-invariant. The dual space is V ∗ = Re∗ where e∗ : V → Rsuch that e∗(e) = 1. Then α∨ = 2e∗ and 〈α, α∨〉 = (2e∗)(e) = 2. R∨ = {α∨ = 2e∗,−2e∗}is a root system in V ∗, which is the dual root system of R. Observe that if we identify Vand V ∗ via e↔ e∗, then α∨ = 2α

(α|α) . See Figure 1.

−e e

Figure 1: Root system for A1, Example 1.8

Example 1.9. (A1-non-reduced): V = Re. The root system is

R = {e, 2e,−e,−2e}.

The dual space is R∗ = Re∗, and the dual root system is R∨ = {±e∗,±2e∗}. E∨ =2e∗, (2e)∨ = e∗. See Figure 2

Remark 1.10. Example 1.8 and Example 1.9 are the only dimension 1 root systems forV = R.

Example 1.11. (A1 ×A1): V = R2 = Re1 ⊕ Re2. The root system is

R = {α = e1,−α, β = e2,−β}.

The dual space is V ∗ = Re∗1 ⊕ Re∗2. We have α∨ = 2e∗1, β∨ = 2e∗2. The dual root systemis R∨ = {±2e∗1,±2e∗2}. This root system will be called reducible. See Figure 3.

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−e−2e e 2e

Figure 2: Root system for A1-non-reduced, Example 1.9

−e1 α = e1

β = e2

−e2

Figure 3: Root system for A1 ×A1, Example 1.11

Example 1.12. (A2): E = R3, V = {(x1, x2, x3) ∈ E : x1 + x2 + x3 = 0}. The rootsystem is

R = {±(e1 − e2),±(e1 − e3),±(e2 − e3)}.

Moreover,W (R) = S3 = {permutations on e1, e2, e3},

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A(R) = S3 × {1,−1}where −1 maps ei to −ei. See Figure 4.

−α α = e1 − e2

β = e2 − e3 α+ β

−α− β −β

Figure 4: Root system for A2, Example 1.12

Example 1.13. (B2): V = R2 = Re1 ⊕ Re2. The root system is

R = {±e1,±e2,±e1 ± e2}.

Moreover,A(R) = W (R) = (Z/2Z)2 o S2.

See Figure 5.

Example 1.14. (C2) – the dual of (B2): The root system is

R = {±2e1,±2e2,±e1 ± e2}

AndA(R) = W (R) = (Z/2Z)2 o S2.

See Figure 6.

Example 1.15. (BC2) – this is non-reduced (also the unique irreducible non-reduced rootsystem of rank 2): V = R2. The root system is

R = {±e1,±e2,±2e1,±2e2,±e1 ± e2}

andA(R) = W (R) = (Z/2Z)2 o S2.

See Figure 7.

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−e1 e1

β = e2

α = e1 − e2

Figure 5: Root system for B2, Example 1.13

α = e1 − e2

β = 2e2

Figure 6: Root system for C2, Example 1.14

Example 1.16. (G2): E = R3, V = {(x1, x2, x3) ∈ E : x1 + x2 + x3 = 0}. The rootsystem is

R = {±(e1−e2),±(e1−e3),±(e2−e3),±(2e1−e2−e3),±(2e2−e1−e3),±(2e3−e1−e2)}

andA(R) = W (R) = dihedral group of order 12.

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e1 − e2

2e1

2e2

Figure 7: Non-reduced root system for BC2, Example 1.15

See Figure 8.

α = e1 − e2

β = −2e1 + e2 + e3

Figure 8: Root system for G2, Example 1.16

Remark 1.17. The above eight examples comprise of all rank 1 and rank 2 root systems(up to isomorphism). The rank 1 root systems are A1, and non-reduced A1. The rank 2root systems are A1 ×A1, A2, B2

∼= C2, G2, BC2.

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Irreducible Root Systems

Let V be the direct sum of Vi, 1 ≤ i ≤ r. Identify V ∗ with the direct sum of V ∗i , and foreach i, let Ri be a root system in Vi. Then R =

∐iRi is a root system in V whose dual

system is R∨ =∐iR∨i . The canonical bijection R↔ R∨ extends each canonical bijection

Ri ↔ R∨i for each i. We say R is the direct sum of root systems Ri.Let α ∈ Ri. If j 6= i, then ker(α∨) ⊃ Vj . So sα induces identity on Vj , j 6= i. On

the other hand, Rα ⊂ Vi, so sα leaves Vi stable. Then W (R) can be identified withW (R1)× · · · ×W (Rr).

Definition 1.18. A root system R is irreducible if R 6= ∅ and R is not the direct sum oftwo nonempty root systems.

It is easy to check that every root system R in V is the direct sum of a family of (Ri)i∈Iof irreducible root systems. The direct sum is unique up to permutation of the index setI. The Ri are called irreducible components of R.

Definition 1.19. A root system R is reduced if α ∈ R implies 12α 6∈ R. α is called

indivisible root .

Here is the complete list of irreducible, reduced root systems (up to isomorphism).

(I) (Al), l ≥ 1 :E = Rl+1, V = {(α1, · · · , αl+1) :

l+1∑i=1

αi = 0},

R = {±(ei − ej) : 1 ≤ i < j ≤ l + 1},#R = l(l + 1),

W (R) = Sl+1, A(R) =

{W (R), if l = 1,

W (R)× Z/2Z, if l ≥ 2.

(II) (Bl), l ≥ 2 :E = V = Rl,R = {±ei, 1 ≤ i ≤ l;±ei ± ej , 1 ≤ i < j ≤ l},#R = 2l2,

A(R) = W (R) = (Z/2Z)l o Sl.

(III) (Cl), l ≥ 2 :E = V = Rl,R = {±2ei, 1 ≤ i ≤ l;±ei ± ej , 1 ≤ i < j ≤ l},#R = 2l2,

A(R) = W (R) = (Z/2Z)l o Sl.

(IV ) (Dl), l ≥ 3 :E = V = Rl,R = {±ei ± ej , 1 ≤ i < j ≤ l},#R = 2l(l − 1),

W (R) = (Z/2Z)l−1 o Sl,

A(R)/W (R) ∼=

{Z/2Z, if l 6= 4,

S3, if l = 4,

(V ) Exceptional root systems: E6, E7, E8, F4, G2.

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Remark 1.20. The above list will classify split, connected, semisimple linear algebraicgroups over an algebraically closed field (up to isogeny).

Angles between Roots

Let α, β ∈ R. Put 〈α, β∨〉 = n(α, β). Then we have

n(α, α) = 2,

n(−α, β) = n(α,−β) = −n(α, β),

n(α, β) ∈ Z,sβ(α) = α− n(α, β)β,

n(α, β) = n(β∨, α∨).

Let (x|y) be a symmetric bilinear form on V , non-degenerate, invariant under W (R). Then

n(α, β) =2(α|β)

(β|β).

So

n(α, β) = 0

⇔ n(β, α) = 0

⇔ (α, β) = 0

⇔ sα and sβ commute ,

and

(α|β) 6= 0⇒ n(β, α)

n(α, β)=

(β|β)

(α|α).

We can determine possible angles between α and β. Let (x|y) be scalar product,W (R)-invariant and α, β ∈ R. Then

n(α, β)n(β, α) =2(α|β)

(β|β)· 2(β|α)

(α|α)= 4 cos2(α̂, β) ≤ 4.

We list all the possibilities in Table 1.

Corollary 1.21. Let α, β ∈ R. If α = cβ, then c ∈ {±1,±2,±12}.

Corollary 1.22. Let α, β be non-proportional roots. If (α|β) > 0 (i.e., if the angle betweenα and β is strictly acute), then α− β is a root. If (α|β) < 0, then α+ β is a root.

Proof. Without loss of generality we may assume ||α|| ≤ ||β||. If (α|β) > 0, then sβ(α) =α − n(α, β)β ∈ R must be α − β by Table 1 (case 1 is the only possibility). Similarly, if(α|β) < 0, then sβ(α) = α−n(α, β)β ∈ R must be α+β (case 2 is the only possibility).

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case angle between α and β order of sαsβ1 n(α, β) = n(β, α) = 0 π/2 2

2 n(α, β) = n(β, α) = 1 π/3 and ||α|| = ||β|| 3

3 n(α, β) = n(β, α) = −1 2π/3 and ||α|| = ||β|| 3

4 n(α, β) = 1, n(β, α) = 2 π/4 and ||α|| =√

2||β|| 4

5 n(α, β) = −1, n(β, α) = −2 3π/4 and ||α|| =√

2||β|| 4

6 n(α, β) = 1, n(β, α) = 3 π/6 and ||α|| =√

3||β|| 6

7 n(α, β) = −1, n(β, α) = −3 5π/6 and ||α|| =√

3||β|| 6

8 n(α, β) = 2, n(β, α) = 2 α = β

9 n(α, β) = −2, n(β, α) = −2 α = −β10 n(α, β) = 1, n(β, α) = 4 β = 2α

11 n(α, β) = −1, n(β, α) = −4 β = −2α

Table 1: Possible Angels Between Two Roots

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2 Review of Algebraic Geometry I (08/26)

The Zariski Topology

Let k be an algebraically closed field (of any characteristic, occasionally char(k) 6= 2, 3).Let V = kn, S = k[T ] := k[T1, T2, · · · , Tn]. f ∈ S can be thought of as a functionf : V → k, via evaluation. We say v ∈ V is a zero of f ∈ k[T ] if f(v) = 0. Wesay v ∈ V is a zero of an ideal I of S if f(v) = 0,∀f ∈ I. Given an ideal I, writeν(I) = set of zeros of I. In the opposite direction, if X ⊂ V , define I(X) ⊂ S = k[T ] tobe the ideal consisting of polynomials f ∈ S with f(v) = 0, ∀v ∈ X.

Example 2.1. Let S = k[T ] = [T1], consider I = (T 2), then ν(I) = {0} and I({0}) = (T ).

Definition 2.2. The radical or nilradical√I of an ideal I is

√I = {f ∈ S : fm ∈ I for some m ≥ 1}.

Theorem 2.3 (Hilbert’s Nullstellensatz). (i) If I is a proper ideal in S, then ν(I) 6= ∅.(ii) For any ideal I of S we have I(ν(I)) =

√I.

Definition 2.4. Observe that(i) ν({0}) = V , ν(S) = ∅;(ii) I ⊂ J ⇒ ν(J) ⊂ ν(I);(iii) ν(I ∩ J) = ν(I) ∪ ν(J);(iv) If (Iα)α∈A is a family of ideals and I =

∑α∈A Iα, then ν(I) =

⋂α∈A ν(Iα).

Note that (i), (ii), (iv) imply that there is a topology on V = kn whose closed setsare the ν(I) where I is an ideal in S – we call it the Zariski topology . A closed subset inthe Zariski topology is called an algebraic set . Also, for any X ⊂ V , we have a Zariskisubspace topology on X.

Proposition 2.5. Let X ⊂ V be an algebraic set.(i) The Zariski topology on X is T1, i.e., points are closed.(ii) The topology space X is noetherian, i.e., it satisfies the following two equivalent

properties: any family of closed subsets of X contains a minimal one , or equivalently ifX1 ⊃ X2 ⊃ X3 ⊃ · · · is a decreasing sequence of closed subsets of X, then there existssome index h such that Xi = Xh for i ≥ h.

(iii) X is quasi-compact, i.e., any open covering of X has a finite subcover.

Note that in algebraic geometry, compact means quasi-compact and Hausdorff.

Review of Reducibility of Topological Spaces

Definition 2.6. A non-empty topological space X is called reducible if it is the union oftwo proper, closed subsets. Otherwise, it is called irreducible.

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Remark 2.7. If X is irreducible, then any two non-empty open subsets of X have anon-empty intersection.

This is mostly interesting only in non-Hausdorff space. In fact, any irreducible Haus-dorff space is simply a point.

If X,Y are two topological spaces. Then

A ⊂ X irreducible ⇔ A is irreducible,

f : X → Y continuous and X irreducible ⇒ f(X) is irreducible.

If X is noetherian topological space, then X has finitely many maximal irreduciblesubsets, called the (irreducible) components of X. The components are closed and theycover X.

Now, we consider the Zariski topology on V = kn.

Proposition 2.8. A closed subset X of V is irreducible if and only if I(X) is prime.

Proof. Let f, g ∈ S with fg ∈ I(X). Then

X = (X ∩ ν(fS)) ∪ (X ∩ ν(gS))

where both X ∩ ν(fS) and X ∩ ν(gS) are closed subsets of V . Since X is irreducible,X ⊂ ν(fS) or X ⊂ ν(gS). Hence f ∈ I(X) or g ∈ I(X). So I(X) is prime.

Conversely, assume I(X) is a prime ideal. If X = ν(I1) ∪ ν(I2) = ν(I1 ∩ I2) andX 6= ν(I1), then there exists f ∈ I1 such that f ∈ I(X). But fg ∈ I(X) for all g ∈ I2.By primeness, g ∈ I(X) implies I2 ⊂ I(X). Hence X = ν(I2). So X is irreducible.

Recall that a topological space is connected if it is not the union of two disjoint properclosed subsets. So if a topological space is irreducible, then it must be connected (but theinverse direction is not true, see Example 2.9).

A noetherian topological space X is a disjoint union of finitely many connected closedsubsets – its connected components. A connected component is a union of irreduciblecomponents. A closed subset X of V = kn is not connected if and only if there exists twoideals I1, I2 of S with I1 + I2 = S and I1 ∩ I2 = I(X).

Example 2.9. X = {(x, y) ∈ k2 : xy = 0} is closed in k2 which is connected, but notirreducible.

Review of Affine Algebras

Let X ⊂ V = kn be an algebraic set. Define

k[X] := {f |X : f ∈ S = k[T ]}.

Then k[X] ∼= k[T ]/I(X) (this is an isomorphism of k-algebra). k[X] is called an affinek-algebra, i.e., it has the following two properties: (i) k[X] is an algebra of finite type, i.e.,

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there exists a finite subset {f1, · · · , fr} of k[X] such that k[X] = k[f1, · · · , fr]; (ii) k[X] isreduced, i.e., 0 is the only nilpotent element of k[X].

An affine k-algebra A also determines an algebraic subset X of some kr such that

A ∼= k[X]. If A ∼= k[T1, · · · , Tr]/I where I = ker(Ti1≤i≤r−−−−→ fi), then

A is reduced ⇔ I is a radical ideal.

The affine k-algebra k[X] determines both the algebraic set X and its Zariski topology.We have the following one-to-one correspondence

{points of X} ↔ Max(k[X]) = {maximal ideals of S containing I(X)}x 7→Mx = IX({x}),

where for Y ⊂ X, IX(Y ) = {f ∈ k[X] : f(y) = 0,∀y ∈ Y }. Note that k[X]/Mx∼= k, so

Mx is a maximal ideal. It is easy to check that(i) x 7→Mx is a bijection;(ii) x ∈ νX(I)⇔ I ⊂Mx;(iii) The closed sets of X are the νX(I), where I is an ideal in k[X];Hence the algebra k[X] determines X and its Zariski topology.

For f ∈ k[X], setDX(f) = D(f) := {x ∈ X : f(x) 6= 0}.

This is an open set of X and we call it a principal open subset of X. It is easy to checkthat the principal opens form a basis for the Zariski topology.

Review of Field of Definitions and F -structures

Definition 2.10. Let F be a subfield of k. We say F is a field of definition of the closedsubset X of V = kn if the ideal I(X) is generated by polynomials with coefficients in F .

SetF [X] := F [T ]/(I(X) ∩ F [T ]).

Then F [T ] ↪→ k[T ] = S induces an isomorphism of F -algebras

F [X] ∼= (an F − subalgebra of S)

and an isomorphism of k-algebras

k ⊗F F [X] ∼= k[X]

(F [X] will be called an F -structure on X). However, this definition of field of definitionand F -structure is not intrinsic.

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Definition 2.11. Let A = k[X] be an affine algebra. An F -structure on X is an F -subalgebra A0 of A which is of finite type over F such that the homomorphism

k ⊗F A0 → A = k[X]

induced by multiplication is an isomorphism. We then write A0 = F [X] and X(F ) :={F − homomorphism : F [X] → F} which is called the F -rational points for the givenF -structure.

Example 2.12. Let k = C and F = R. Let X = {(z, w) ∈ C2 : z2 +w2 = 1}, A = k[X] =C[T,U ]/(T 2 + U2 − 1). Let a = T mod (T 2 + U2 − 1), b = U mod (T 2 + U2 − 1). Hereare two R-structure on X:

A1 = R[a, b],

A2 = R[ia, ib].

These are two different R-structures. To see this, consider the R-rational points for A1

and A2. The R-rational points for A1 is

X(R) = {R− homomorphism R[a, b]→ R} = S1

while the R-rational points for A2 is

X(R) = {R− homomorphism R[ia, ib]→ R} = ∅.

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3 Review of Algebraic Geometry II, Introduction to LinearAlgebraic Groups I (09/02)

Review of Regular Functions

Let x ∈ X ⊂ V = kn.

Definition 3.1. A function f : U → k with U a neighborhood of x in X is regular at x if

f(y) =g(y)

h(y), g, h ∈ k[X]

on a neighborhood V ⊂ U ∩D(h) of x (i.e., h 6= 0 in V ). As usual, we say f is regular ina non-empty, open subset U if it is regular at each x ∈ U . We define

OX(U) = O(U) := the k − algebra of regular functions in U.

Observe that if U, V are non-empty, open sets and U ⊂ V , then the restriction O(V )→O(U) is a k-algebra homomorphism.

Let U =⋃α∈A Uα be an open cover of the open set U . Assume that for each α, we

have fα ∈ O(Uα) such that if Uα∩Uβ 6= ∅, then fα and fβ restrict to the same function inO(Uα ∩ Uβ). Then there exists f ∈ O(U) such that f |Uα = fα for any α ∈ A (patching).(X,O) is called a ringed space and O is called a sheaf of k-valued functions on X.

Definition 3.2. The ringed space (X,OX) (or simply X) as above is called an affinealgebraic variety over k or an affine k-variety or simply an affine algebraic variety.

Lemma 3.3. Let (X,OX) be an affine algebraic variety. Then the homomorphism

ϕ : k[X]→ O(X)

f 7→ f/1

is an isomorphism of k-algebras.

If (X,OX) and (Y,OY ) are two ringed space or affine algebraic varieties, and φ : X → Yis a continuous map, and f is a function on an open set V ⊂ Y , then define

φ∗V (f) := f ◦ φ|φ−1(V ),

a function on an open subset φ−1(V ) ⊂ X.

Definition 3.4. φ is called a morphism of ringed space or of affine algebraic varieties iffor each V ⊂ Y , φ∗V maps OY (V ) into OX(φ−1V ).

17

If X ⊂ Y , φ : X ↪→ Y is injection and OX = OY |X , then φ : X ↪→ Y is a morphism ofringed spaces. This is the notion of ringed subspace.

A morphism ϕ : X → Y of affine algebraic varieties induces an algebraic homomor-phism OY (Y )→ OX(X) by composition with ϕ. Then we get an algebraic homomorphismϕ∗ : k[Y ]→ k[X] by Lemma 3.3. Conversely, an algebraic homomorphism ψ : k[Y ]→ k[X]also gives a continuous map (ψ) : X → Y such that (ψ)∗ = ψ. Hence there is an equiva-lence of categories{

affine k-varieties and their morphisms}←→ {affine k-algebras and their homomorphisms}.

Let F be a subfield of k. Similar remarks apply to affine F -varieties and F -subalgebras.Hence affine F -varieties can also be described algebraically. An example is that the affinen-space An, n ≥ 0 with algebra k[T1, T2, · · · , Tn].

Review on Products

Given two affine algebraic varieties X and Y over k, we would like to define a productaffine algebraic variety X × Y .

Definition 3.5 (Universal Property of Product (in any category)). A product of X and Yis defined as an affine algebraic variety Z together with morphisms p : Z → X, q : Z → Ysuch that the following holds: for any triaple (Z ′, p′, q′) as above, there exists a uniquemorphism r : Z ′ → Z such that the diagram

Z ′

X <p

p′

<Z

r

∨

..........q> Y

q′

>

commutes.

Equivalently, we can do this in the category of affine k-algebras. Put A = k[X],B = k[Y ], and C = k[Z]. Then using the equivalence of categories we can expressthe universal property algebraically: there exists k-algebra homomorphisms a : A → C,b : B → C such that for any triple (C ′, a′, b′) of affine k-algebras, there is a unique k-algebrahomomorphism c : C → C ′ such that the diagram

C

Aa′>

a>

C ′

c∨

.........

<b′

B

b

<

commutes.

18

Having this property just for the k-algebras (forgetting that C is an affine k-algebra)we already know from abstract algebra that C = A⊗k B with

a(x) = x⊗ 1, x ∈ A,b(y) = 1⊗ y, y ∈ A,

satisfies all the requirements.

Lemma 3.6. Let A, B be k-algebras of finite type.(i) If A,B are reduced, then A⊗k B is reduced.(ii) If A,B are integral domains, then A⊗k B is an integral domain.

Therefore, for X,Y affine k-varieties, a product variety X × Y exists (as an affinek-variety). It is unique up to isomorphism. If X and Y are irreducible, then so is X × Y .In fact, it is easy to see the set underlying X × Y can be identified with the product ofthe sets underlying X and Y . With this identification, the Zariski topology on X × Y isfiner than the product topology. If F is a subfield of k, a product of two affine F -varietiesexists and is unique up to F -isomorphism.

Prevarieties and Varieties

Definition 3.7. A prevariety over k is a quasi-compact ringed space (X,O) such thatany point of X has an open neighborhood U such that

(U,O|U ) ∼= an affine k-variety

is an isomorphism in the category of affine k-algebras or affine k-varieties.

Definition 3.8. A morphism of prevarieties is a morphism of ringed spaces.

Definition 3.9. A sub prevariety of a prevariety is a ringed subspace which is also aprevariety.

A product of two prevarieties exists and is unique up to isomorphism. This allows usto consider the diagonal subset ∆X = {(x, x) : x ∈ X} of X×X equipped with its reducedtopology. Denote by

i : X → ∆X

x 7→ (x, x).

Then i : X → ∆X is a homeomorphism of topological spaces for any prevariety X.

Definition 3.10. A prevariety X is called a variety or an algebraic variety over k ork-variety if it satisfies the Separation Axiom, i.e.,

(Separation Axiom): ∆X is closed in X ×X.

19

Morphisms of varieties are now defined in the usual way.

Example 3.11. Let X be an affine k-variety. Then ∆X = νX×X(I) where I is the kernelof the map defined from universal property

k[X ×X] = k[X]⊗k k[X]→ k[X].

In fact, I is generated by f ⊗1−1⊗f , f ∈ k[X]. Hence X satisfies the Separation Axiom,i.e., it is a variety over k. Also note that

k[X ×X]/I ∼= k[X],

which implies that i gives a homeomorphism of topological spaces X → ∆X .

Lemma 3.12. A topological space X is Hausdorff if and only if ∆X is closed in X ×Xfor the product topology.

Lemma 3.13. The product of two varieties is a variety.

Lemma 3.14. For X a variety, Y a prevariety, if ϕ : Y → X is a morphism of prevari-eties, then its graph Γφ = {(y, φ(y)) : y ∈ Y } is closed in Y ×X.

Lemma 3.15. Again, for X a variety, Y a prevariety, if two morphisms ϕ : Y → X,ψ : Y → X coincide on a dense subset, then ϕ = ψ.

Lemma 3.16 (Criterion for a prevariety to be a variety). (i) Let X be a variety, U, V beaffine open sets in X. Then U ∩ V is an affine open set and the images under restrictionof OX(U) and OX(V ) in OX(U ∩ V ) generate it.

(ii) Let X be a prevariety and let X = ∪mi=1Ui be a covering by affine open sets. ThenX is a variety if and only if for each pair (i, j), the intersection Ui ∩ Uj is an affine openset and the images under restriction of OX(Ui) and OX(Uj) in OX(Ui ∩Uj) generate it.

Remark 3.17. There are more examples of varieties, for example, projective varieties,which are not affine.

Definition of Linear Algebraic Groups

Now we introduce the notion of linear algebraic groups.

Definition 3.18. Let k be an algebraically closed field, and let F be a subfield. Analgebraic group G is an algebraic variety over k which is also a group such that the maps

µ : G×G→ G

(x, y) 7→ xy

and

i : G→ G

x 7→ x−1

are morphisms of varieties. An algebraic group G is called a linear algebraic group if it isaffine as an algebraic variety.

20

Definition 3.19. Let G, G′ be algebraic groups. A homomorphism of algebraic groupsϕ : G → G′ is a group homomorphism and a morphism of varieties. (Hence we have thenotion of isomorphism and automorphism of algebraic groups).

Note that G × G′ is automatically an algebraic group – called the direct product ofG×G′.

A closed subgroup H of an algebraic group G (with respect to the Zariski topology)can be made into an algebraic group such that H ↪→ G is a homomorphism of algebraicgroups.

Definition 3.20. The algebraic group G is called an F -group where F ⊂ k is a subfield if(i) G is an F -variety;(ii) the morphisms µ and i are defined over F ;(iii) the identity element e is an F -rational point.

Similarly, we get F -homomorphisms. For G an F -group, set

G(F ) := the set of F -rational points, which come with a canonical group structure.

Let G be a linear algebraic group. Put A = k[G]. Recall that there is an equivalenceof categories{

affine k-varieties and their morphisms}←→ {affine k-algebras and their homomorphisms}.

So the morphisms µ and i can be described as algebraic homomorphisms. µ is definedby ∆ : A → A ⊗k A, called “multiplication”. i can be defined by ι : A → A, called“amtipode”. Moreover, the identity element e is a homomorphism A → k. With this inhand, we can write the group axioms algebraically. We denote

m : A⊗k A→ A

f ⊗ g 7→ fg

and

ε : Aε> A

k.∪

∧e >

Then associativity in Group Axioms is the same as the diagram

A∆

> A⊗k A

A⊗k A

∆

∨∆⊗kid

> A⊗k A⊗k A

id⊗∆

∨

21

commutes. The existence of the inverse in Group Axioms is the same as the diagram

A⊗k Ai⊗id> A⊗k A

A

∆

∧

ε> A

m

∨

A⊗k A

∆

∨id⊗i> A⊗k A

m

∧

commutes. The existence of identity in Group Axioms is the same as the diagram

A <e⊗id

A⊗k A

A⊗k A

id⊗e∧

<∆

A

∆

∧id

<

commutes.

22

4 Introduction to Linear Algebraic Groups II (09/09)

Examples of Algebraic Groups

We first give several examples of algebraic groups.Recall that k is algebraically closed, and F ⊂ k is a subfield.

Example 4.1. G = k = A1. Another notation is Ga – “the additive group”. A = k[G] =k[T ]. Multiplication and inversion are

∆ : k[T ]→ k[T ]⊗k k[T ] ∼= k[T,U ]

T 7→ T + U

and

∆ : k[T ]→ k[T ]

T 7→ −T.

Note that G is a variety because we have the separation axiom: ∆G = {(g, g) : g ∈ G} isclosed in G×G. Therefore, ∆ and ι are k-algebra homomorphism. This implies µ, i givenby

µ(x, y) = x+ y, i(x) = −x,are morphisms of varieties. For any F ⊂ k, F [T ] defines an F -structure on Ga:

Ga(F ) ∼= F.

Example 4.2. G = k∗ = A1\{0}. Other notation for this group is Gm – “the multiplica-tive group”, or GL1. A = k[G] = k[T, T−1]. Multiplication and inversion are

∆ : k[T, T−1]→ k[T, T−1]⊗k k[T, T−1] ∼= k[T, T−1, U, U−1]

T 7→ TU

and

ι : k[T, T−1]→ k[T, T−1]

T 7→ T−1.

Also,

e : k[T, T−1]→ k

T 7→ 1

Again, F [T, T−1] defines an F -structure, Gm(F ) ∼= F ∗. Observe that for any n ∈ Z\{0},x 7→ xn defines a homomorphism of algebraic groups Gm → Gm. When is this anisomorphism?

Gm → Gm is an isomorphism⇔ φ∗ : A = k[T, T−1]→ A is an isomorphism.

Hence Aut(Gm) ∼= {±1}.

23

Example 4.3. G = An, n ≥ 1. µ and i are given by

µ(x, y) = xy, i(x) = −x,

and e = 0. In particular, G = Mn = {all n× n matrices} ∼= kn2.

Example 4.4. G = GLn = {x ∈Mn : D(x) 6= 0} where D is the determinant. Note thatD is a regular function on Mn, and GLn is the principal open set given by D 6= 0. µ andi are given by

µ(x, y) = xy, i(x) = x−1,

and e = In. The k-algebra is A = k[GLn] = k[Tij , D−1]1≤i,j≤n,D=det(Tij) with homomor-

phisms

∆ : A→ A⊗k A

Tij 7→n∑h=1

TihThj

ι : A→ A

Tij 7→ (i, j)− entry of the inverse

of the matrix [Tkl]1≤k,l≤n,

and

e : A→ k

Tij 7→ δij .

For any F ⊂ k, F [Tij , D−1] defines an F -structure on G = GLn and G(F ) = GLn(F ).

Note that any Zariski closed subgroup of GLn defines a linear algebraic group.

Example 4.5. Any finite closed subgroup of GLn is a linear algebraic group.

Example 4.6. Dn, the diagonal matrices in GLn, is a linear algebraic group.

Example 4.7. Tn, the upper triangulars in GLn, is a linear algebraic group.

Example 4.8. Un, the unipotent upper triangular matrices in GLn, is a linear algebraicgroup.

Example 4.9. SLn = {X ∈ GLn : det(X) = 1}, the special linear group, is a linearalgebraic group.

Example 4.10. On = {X ∈ GLn : tXX = 1}, the orthogonal group, is a linear algebraicgroup. Let

J =

1...

1

.

Then On = On(J) = {X ∈ GLn : tXJX = J}.

24

Example 4.11. SOn = On ∩ SOn, the special orthogonal group, is a linear algebraicgroup. Sometimes we distinguish the odd and the even indices as SO2n+1 and SO2n.

Example 4.12. The special orthogonal group, Sp2n = {X ∈ GL2n : tXJX = J} where

J =

1...

1−1

...

−1

2n×2n

or J =

(0 In−In 0

),

is a linear algebraic group.

Review of Projective Varieties

Definition 4.13. The projective space Pn is the set {1−dim subspace of kn+1} or equiv-alently

kn+1\{0}/∼

wherex ∼ y ⇐⇒ y = ax for some a ∈ k∗ = k\{0}.

If x = (x0, x1, · · · , xn) ∈ kn+1\{0}, we write x∗ or [x0 : x1 : · · · : xn] for the equivalenceclass of x. The xi’s are called the homogeneous coordinates of x∗.

We cover the set Pn by U0, U1, · · · , Un where

Ui := {(x0, x1, · · · , xn)∗ : xi 6= 0}.

Each Ui can be given an affine variety structure of An via

ϕi : Ui → An

(x0, x1, · · · , xn)∗ 7→(x0

xi,x1

xi, · · · , x̂i

xi, · · · , x0

xi

).

Then ϕi(Ui ∩ Uj) is a principal open D(f) in An because we may take

f =

Tj , j > i

1, j = i

Tj+1, j < .

Declare a subset U of Pn open if U∩Ui is open in the affine variety Ui for any i = 0, 1, · · · , n.For x ∈ Pn, assume x ∈ Ui for some i. Then a function f in a neighborhood of x is declared

25

regular at x if f |Ui is regular in the affine structure of Ui and we get a sheaf OPn and aringed space (Pn,OPn) that makes Pn into a prevariety.

In fact, Pn is a variety. We can check this by using the criterion we had before inLemma 3.16.

Definition 4.14. A projective variety is a closed subvariety of some Pn. A quasi-projectivevariety is an open subvariety of a projective variety.

Closed sets in Pn are of the form

ν∗(I) = {x∗ ∈ Pn : x ∈ νkn+1(I)}

where I is a homogeneous ideal. Recall that a homogeneous ideal means an ideal I ∈ S =k[T0, T1, · · · , Tn] generated by homogeneous polynomials.

Example 4.15. We assume char(k) 6= 2, 3. Define

G = {(x0, x1, x2)∗ ∈ P2 : x0x22 = x3

1 + ax1x20 + bx3

0}

where a, b ∈ k such that the polynomial T 3+aT+b has no multiple roots. Let e = (0, 0, 1)∗

be the point at “∞′′. Define the sum of three corlinear points in P2 to be e. It is easy tocheck that if x = (x0, x1, x2)∗ ∈ G, then −x = (x0, x1,−x2)∗. It is a bit of work to writeaddition explicitly. We can also check the associativity. Then G is an algebraic group,which is non-linear.

Review of Dimension

Let X be an irreducible variety. First, assume X to be affine. Since X is irreducible,k[X] is an integral domain. Then we get its fraction field k(X). It is an easy fact (bylocalization) that if U is any open affine subset of X, then

k(U) ∼= k(X).

If X is any variety, then the above and the criterion for a prevariety to be a variety inLemma 3.16 imply that if U, V are any two affine open sets, then k(U), k(V ) can becanonically identified. Hence we can speak of the fraction field k(X).

Definition 4.16. We define the dimension of an irreducible variety X to be

dimX = transcendence degree of k(X) over k.

If X is reducible and (Xi)1≤i≤m are its irreducible components, then

dimX = max1≤i≤m

dimXi.

26

Lemma 4.17. If X is affine and k[X] = k[x1, x2, · · · , xr], then

dimX = maximal number of elements among x1, · · · , xr that are algebraically independent over k.

Lemma 4.18. If X is irreducible and Y is proper irreducible closed subvariety of X, then

dimY < dimX.

Lemma 4.19. If X,Y are irreducible varieties, then

dim(X × Y ) = dimX + dimY.

Lemma 4.20. If ϕ : X → Y is a morphism of affine varieties and X is irreducible, thenϕ(X) is irreducible, and dimϕ(X) ≤ dimX.

Example 4.21. dim An = n, and dim Pn = n.

Remark 4.22. If U is an open set in X, then dimU = dimX. If dimX = 0, then Xis finite. If f ∈ k[T1, · · · , Tn] is irreducible, then ν(f) is (n − 1)-dimensional irreduciblesubvariety of An. Dimension respects field of definition. In other words, if X is an F -variety, then

dimX = transcendence degree of F (X) over F .

Basic Results on Algebraic Groups

Let k be an algebraically closed field, G an algebraic group. For g ∈ G, the maps

Lg : G→ G

x 7→ gx

and

Rg : G→ G

x 7→ xg

define isomorphisms of the varieties G.

Proposition 4.23. (i) There is a unique irreducible component G0 of G that contains e.It is closed, normal subgroup of finite index.

(ii) G0 is the unique connected component of G containing e.(iii) Any closed subgroup of G of finite index contains G0.

Proof. (i) Let X,Y be two irreducible components of G containing e. Then XY = ν(X ×Y ) is irreducible, and its closure XY is irreducible, closed. But irreducible componentsare maximal irreducible closed subsets, so X = XY = Y . This implies X is closed undermultiplication. Now, i is a homomorphism, hence X−1 is an irreducible component of G

27

containing e. So X−1 = X, i.e., X is a closed subgroup. Now for g ∈ G, gXg−1 is anirreducible component containing e. This implies gXg−1 = X for any g ∈ G. So X isa normal subgroup of G. So gX must be the irreducible components of G and there arefinitely many of them. Hence G0 = X satisfies (i).

(ii) The cosets gG0 are mutually disjoint, and each connected component is a union ofthem. So the irreducible and connected components of G must coincide. This proves (ii).

(iii) Let H be a closed subgroup of G of finite index, then H0 is a closed subgroupof finite index in G0. Now H0 is both open and closed in G0, but G0 is connected, soH0 = G0.

Convention: we talk about “connected algebraic groups” and not “irreducible algebraicgroups”.

We need the following two lemmas about morphisms of varieties.

Lemma 4.24. If φ : X → Y is a morphism of varieties, then φ(X) contains a nonemptyopen subset of its closure φ(X).

Lemma 4.25. If X,Y are F -varieties, and φ is defined over F , then φ(X) is an F -subvariety of Y .

Proposition 4.26. Let φ : G→ G′ be a homomorphism of algebraic groups. Then(i) kerφ is a closed normal subgroup of G.(ii) φ(G) is a closed subgroup of G.(iii) If G and G′ are F -groups and φ is defined over F , then φ(G) is an F -subgroup of G′.(iv) φ(G0) = φ(G)0.

We need the following two lemmas to prove it.

Lemma 4.27. If U and V are dense open subgroups of G, then G = UV .

Lemma 4.28. If H is a subgroup of G, then(i) The closure H is also a subgroup of G.(ii) If H contains a non-empty open subset of H, then H = H.

Proposition 4.29 (Chevalley). Let (Xi, φi)i∈I be a family of irreducible varieties andmorphisms φi : Xi → G. Denote by H the smallest closed subgroup of G containingYi = φi(Xi). Assume that all Yi contain e. Then(i) H is connected.(ii) H = Y ±1

i1Y ±1i2· · ·Y ±1

infor some n ≥ 0, i1, · · · , in ∈ I.

(iii) If G is an F -group, and for all i ∈ I, Xi is an F -variety, and φi is defined over F ,then H is an F -subgroup of G.

Corollary 4.30. (i) If H and K are closed subgroups of G, one of which is connected,then the commutator subgroup (H,K) is connected.

(ii) If G is an F -group and H,K are F -subgroups, then (H,K) is a connected F -subgroup. In particular, (G,G) is a connected F -subgroup.

28

5 Introduction to Linear Algebraic Groups III (09/16)

G-spaces

Let k be an algebraically closed field, X an variety over k, G an algebraic group over k.

Definition 5.1. Let a : G×X → X defined by a(g, x) = g · x be a morphism of varietiessuch that

g · (h · x) = (gh) · x, ∀g, h ∈ G,e · x = x.

Then X is called a G-space or G-variety .

Definition 5.2. Let F ⊂ k be a subfield. If G is an F -group and X is an F -variety, anda is defined over F , then we say X is a G-space over F .

Definition 5.3. If F acts trivially on the G-space X, we say X is a homogeneous spacefor G. For x ∈ X, define the orbit of X to be

G · x = {g · x : g ∈ G}

and the isotropy group of x to be

Gx = {g ∈ G : g · x = x}.

Lemma 5.4. Gx is a closed subgroup of G.

Proof. Fix x ∈ X.

G→ G×X → X

g 7→ (g, x) 7→ g · xis continuous and Gx is the inverse image of {x}, and {x} is closed in the Zariski topology,so Gx is closed.

Definition 5.5. Let X and Y be G-spaces. A morphism ϕ : X → Y is called a G-morphism or G-equivalent if

ϕ(g · x) = g · ϕ(x), ∀g ∈ G, x ∈ X.

Lemma 5.6. (i) An orbit G · x is open in G · x.(ii) There exists closed orbits.

Proof. (i) Fix x ∈ X and consider the morphism ϕ : G → X given by ϕ(g) = g · x. Bya general fact from algebraic geometry, we know ϕ(G) = G · x contains a nonempty opensubset U in its closure ϕ(G) = G · x. Now G · x =

⋃g∈G g · U , so G · x is open in G · x.

(ii) Let Sx = G · x−G · x, which is closed in X. It is a union of orbits. Consider thefamily {Sx}x∈X of closed subsets in X. It has a minimal subset Sx0 . By (i), Sx0 must beempty. Then G · x = G · x is closed.

Corollary 5.7. G · x is locally closed in X, i.e., an open subset of a closed set in X. Ithas an algebraic variety structure, and is automatically a homogeneous space for G.

29

Examples of G-spaces

Example 5.8 (Inner automorphisms). X = G, a : G × G → G is defined by a(g, x) =gxg−1. The orbits are conjugacy classes G · x = {gxg−1 : g ∈ G}. The isotropy group isGx = CG(x) = {g ∈ G : gx = xg}.

Example 5.9 (Left and right actions). X = G, a : G×G→ G is defined by (g, x) 7→ gx or(g, x) 7→ xg−1. G acts simply-transitively, i.e., Gx = {1} ∀x ∈ G, and G is a homogeneousspace. Then G is called a principal homogeneous space.

Example 5.10. Let V be a finite dimensional vector space over k of dimension n. Arational representation of G in V is a homomorphism of algebraic groups r : G→ GL(V ).V is also called a G-module, via g · v = r(g)v.

Remark 5.11. Let F ⊂ k be a subfield. View V as a finite dimensional vector space withan F -structure and view GL(V ) as an F -group and r is defined over F , then we call r arational map over F .

Example 5.12. With the same notation, any closed subgroup G of GLn acts on X = An

(left action) so An is a G-space. The orbits of X are {0} and An\{0}. For example, forG = SLn, the orbit is {0}.

Now assume G is affine. X is an affine G-space with action a : G×X → X. We havek[G×X] = k[G]⊗k k[X] and a is given by a∗ : k[X]→ k[G]⊗k k[X]. For g ∈ G, x ∈ X,f ∈ k[X], define

s(g) : k[X]→ k[X]

(s(g)f)(x) = f(g−1x).

Then s(g) is an invertible linear map from (often infinite-dimensional) vector space k[X]to itself. This way, we get a representation of abstract groups s : G→ GL(k[X]).

Proposition 5.13. Let V be a finite dimensional subspace of k[X].(i) There is a finite dimensional subspace W of k[X] containing V such that s(g)W ⊂

W , ∀g ∈ G.(ii) V is stable under all s(g) if and only if a∗(V ) ⊂ k[G]⊗k V . In this case, we get a

map sV : G× V → V which is a rational representation of G in V .(iii) If G is an F -group, X is an F -variety, V is defined over F , and a is an F -

morphism, then W in part (i) can be taken to be defined over F .

Proof. (i) Without loss of generality, we may assume that V = kf is one dimensional.Write

a∗(f) =n∑i=1

ui ⊗ fi, ui ∈ k[G], fi ∈ k[X].

30

Then (s(g)f)(x) = f(g−1x) =∑n

i=1 ui(g−1)fi(x). Now W ′ = 〈fi〉i=1,··· ,n is finite dimen-

sional and let W be its subspace spanned by all s(g)f , g ∈ G. Then W satisfies (i).(ii) (⇐) is just as in (i).(⇒) Assume V is s(G)-stable. Let (fi) be a basis for V and extend it to a basis

(fi) ∪ (gi) for k[X]. Take f ∈ V , and write

a∗(f) =∑i

ui ⊗ fi +∑j

vj ⊗ gj , ui, vj ∈ k[G].

Nows(g)f =

∑i

ui(g−1)fi +

∑j

vj(g−1)gj .

By assumption, vj(g−1) = 0 for all g ∈ G. Hence vj = 0 for all j. So a∗f ∈ k[G]⊗k V .

(iii) In the argument for (i), check that if all data is defined over F , then so is W .

Observe that there exists an increasing sequence of finite dimensional subspaces (Vi)of k[X] such that (i) each Vi is stable under s(G) and s defines a rational map of G in Vi,and (ii) k[X] =

⋃i Vi.

Now we still assume that G is affine. Consider the left and right action of G on itself.For g, x ∈ G, f ∈ k[G], define

(λ(g)f)(x) = f(g−1x),

(ρ(g)f)(x) = f(xg).

They both define representations of abstract group G in GL(k[G]). If ι : k[G] → k[G] isthe automorphism of k[G] defined by inversion in G, then we have

ρ = ι ◦ λ ◦ ι−1.

Lemma 5.14. Both λ and ρ have trivial kernels, i.e., they are “faithful” representations.

Proof. If λ(g) = id, then f(g−1) = f(e) for all f ∈ k[G]. Hence g−1 = e. So g = e. Thisproves that kerλ is trivial. The proof for ρ is similar.

Theorem 5.15. Let G be a linear algebraic group.(i) There is an isomorphism of G onto a closed subgroup of some GLn.(ii) If G is an F -group, the isomorphism may be taken to be defined over F .

Proof. (i) By part (i) of Proposition 5.13, we may assume k[G] = k[f1, · · · , fn] where (fi)is a basis of ρ(G)-stable subspace V of k[G]. By part (ii) of Proposition 5.13, we can write

ρ(g)fi =n∑j=1

mji(g)fj , mji ∈ k[G],∀g ∈ G, i, j = 1, · · · , n.

31

Define

φ : G→ GLn

g 7→ (mij(g))n×n.

Then φ is a group homomorphism and a morphism of affine varieties.We claim that φ is injective. If φ(g) = e, then ρ(g)fi = fi, ∀i. But ρ(g) is an algebraic

homomorphism and k[G] is generated by the fi, so

ρ(g)f = f, ∀f ∈ k[G].

Hence g = e.We claim that φ∗ is surjective. Note that φ∗ : k[GLn] = k[Tij , D

−1]→ k[G] is given by

φ∗(Tij) = mij ,

φ∗(D−1) = det(mij)−1.

But fi(g) =∑

jmji(e)fj(e), so each fi is in im(φ∗), hence φ∗ is surjective. This impliesthat φ(G) is a closed subgroup of GLn. Its algebra is isomorphic to k[GLn]/kerφ∗ ∼= k[G].Therefore, φ is an isomorphism of algebraic groups G ∼= φ(G). So we have proved (i).

For (ii), we check that the maps above can be taken to be defined over F .

Lemma 5.16. Let H be a closed subgroup of G. Then

H = {g ∈ G : λ(g)IG(H) = IG(H)} = {g ∈ G : ρ(g)IG(H) = IG(H)}.

Proof. We consider λ. The proof for ρ is similar. For g, h ∈ H, f ∈ IG(H), we have(ρ(g)f)(h) = f(g−1h) = 0, so ρ(g)f ∈ IG(H). This proves H ⊂ {g ∈ G : λ(g)IG(H) =IG(H)}. Now assume that g ∈ G and ρ(g)IG(H) = IG(H). Then for all f ∈ IG(H) wehave f(g−1) = (λ(g)f)(e) = 0. So g−1 ∈ H, and hence g ∈ H. This proves H ⊃ {g ∈ G :λ(g)IG(H) = IG(H)}.

32

6 Jordan Decomposition (09/23)

Jordan Decomposition

Definition 6.1. Let V be a finite dimensional vector space over an algebraically closedfield k. Let x ∈ End(V ). x is called nilpotent if xn = 0 for some n ≥ 1 ( ⇐⇒ 0 is theonly eigenvalue of x). x is semisimple if the minimal polynomial of x has distinct roots(⇐⇒ x is diagonalizable over k). x is unipotent if x = 1 + n where n is nilpotent.

Remark 6.2. 0 is the only endomorphism of V that is both nilpotent and semisimple.

Proposition 6.3 (Additive Jordan Decomposition). Let x ∈ End(V ).(i) There exists unique xs, xn ∈ End(V ) such that x = xs + xn and xs is semisimple,

xn is nilpotent, and xs · xn = xn · xs.(ii) There exists polynomials p(T ), q(T ) ∈ k[T ] satisfying p(0) = q(0) = 0 such that

xs = p(x) and xn = q(x). In particular, xs and xn commute with x and in fact, theycommute with any endomorphism of V that commutes with x.

(iii) If A ⊂ B ⊂ V are subspaces and x(B) ⊂ A, then xs(B) ⊂ A, xn(B) ⊂ A.(iv) If xy = yx for some y ∈ End(V ), then

(x+ y)s = xs + ys,

(x+ y)n = xn + yn.

Corollary 6.4 (Multiplicative Jordan Decomposition). Let x ∈ GL(V ). There existsunique elements xs, xu ∈ GL(V ) such that x = xsxu = xuxs and xs is semisimple, xu isunipotent.

Remark 6.5. Suppose V is a finite dimensional vector space over an algebraically closedfield k. Let a ∈ End(V ). Let W ⊂ V be a a-stable space. Then W is stable under asand an and a|W = as|W + an|W and a = as + au where¯means the linear transformationinduced on V/W . Similarly, if a ∈ GL(V ), then a|W = as|W · au|W , and similarly forV/W .

Remark 6.6. Suppose V,W are two finite dimensional vector space over k. Let ϕ : V →W be linear. Let a ∈ End(W ), b ∈ End(W ). If ϕ ◦ a = b ◦ ϕ, i.e., the diagram

Va> V

W

ϕ

∨b> W

ϕ

∨

is commutative, thenϕ ◦ as = bs ◦ ϕ

33

andϕ ◦ an = bn ◦ ϕ.

Let V be a not necessarily finite dimensional vector space over k. Again

End(V ) := algebra of endomorphisms of V,

GL(V ) := group of invertible endomorphisms of V.

We say a ∈ End(V ) is locally finite if V is a union of finite dimensional a-stable subspaces.We say a ∈ End(V ) is semisimple if its restriction to any finite dimensional a-stablesubspace is semisimple. We say a ∈ End(V ) is locally nilpotent if its restriction to anyfinite dimensional a-stable subspace is nilpotent. We say a ∈ End(V ) is locally unipotentif its restriction to any finite dimensional a-stable subspace is unipotent. For a locallyfinite a ∈ End(V ), we have a = as + an with as locally finite and semisimple, an locallyfinite and locally nilpotent. For x ∈ V , take a finite dimensional a-stable subspace Wcontaining x, and put

asx := (a|W )s,

anx := (a|W )n.

It follows from Remark 6.5 that asx and anx are independent of the choice of W . Ifa ∈ GL(V ), we have a similar multiplicative Jordan decomposition a = as · au where as issemisimple, au is locally unipotent.

Remark 6.7. There is an infinite-dimensional generalization of Remark 6.6.

Jordan Decomposition in Linear Algebraic Groups

We now come to the Jordan decomposition in linear algebraic groups.Let G be a linear algebraic group and A = k[G]. From our discussion of G-actions, we

can conclude that the right translation ρ(g), g ∈ G, is a locally finite element of GL(A),i.e., ρ(g) = ρ(g)sρ(g)u.

Theorem 6.8. (i) There exists unique elements gs and gu in G such that ρ(g)s = ρ(gs),ρ(g)u = ρ(gu), and g = gsgu = gugs.

(ii) If φ : G → G′ is a homomorphism of linear algebraic groups, then φ(g)s = φ(gs)and φ(g)u = φ(gu).

(iii) If G = GLn, then gs and gu are the semisimple and unipotent parts of g ∈ GL(V ),where V = kn as before.

Remark 6.9. gs is called the semisimple part of g, and gu is called the unipotent part ofg.

34

Proof of Theorem 6.8. (i) Let m : A ⊗ A → A be the k-algebra homomorphism corre-sponding to multiplication in G. ρ(g) is an algebra automorphism of A. That means

m ◦ (ρ(g)⊗ ρ(g)) = ρ(g) ◦m.

By Remark 6.7, we have

m ◦ (ρ(g)s ⊗ ρ(g)s) = ρ(g)s ◦m.

So ρ(g)s is also an automorphism of A. So f 7→ (ρ(g)sf)(e) defines a homomorphismA → k, i.e., a point in G, and we call it gs. Now ρ(g) commutes with all left translationλ(x), x ∈ G, and the λ(x) are locally finite, so ρ(g)s also commutes with all λ(x). In otherwords, for f ∈ A,

(ρ(g)sf)(x) = (λ(x−1)ρ(g)sf)(e)

= (ρ(g)sλ(x−1)f)(e)

= (λ(x−1)f)(gs) (by definiton of gs)

= f(xgs)

= (ρ(gs)f)(x).

Hence ρ(g)s = ρ(gs). A similar argument also gives ρ(g)u = ρ(gu). So

ρ(g) = ρ(g)sρ(g)u = ρ(gs)ρ(gu) = ρ(gsgu).

But ρ is a faithful representation of G (i.e., ker ρ is trivial), so g = gsgu. Similarly g = gugs.(ii) Recall that for homomorphism of algebraic groups φ : G → G′, we saw that

Im(φ) = φ(G) is closed in G′. So φ can be factored into

G→ Im(φ)→ G′.

This reduces the proof to two cases: case (a) the inclusion Imφ → G′, and case (b) thesurjection G→ Imφ.

For case (a), G is a closed subgroup of G′ and φ is the inclusion. Let k[G] = k[G′]/I.By Lemma 5.16,

G = {g ∈ G′ : ρ(g)I = I}.

Now W = I is a subspace of V = k[G′] and it is stable under ρ(g), so by Remark 6.5, wehave a Jordan decomposition on V/W = k[G]. So

φ(g)s = φ(gs),

φ(g)u = φ(gu),

as φ is just inclusion.

35

For case (b), if φ is surjective, then k[G′] can be viewed as a subspace of k[G], whichis stable under all ρ(g), g ∈ G. Again, the result follows from Remark 6.5.

(iii) Let G = GL(V ) with V = kn. Let 0 6= f ∈ V ∨ = dual of V and define f̃(v) ∈ k[G]via f̃(v)(g) = f(gv). Then f̃ is an injective linear map V → k[G], and ∀x ∈ G, we have

f̃(gv)(x) = f(xgv) = f̃(v)(xg) = [ρ(g)f̃(v)](x).

Hence f̃(gv) = ρ(g)f̃(v). By Remark 6.7, we have

f̃(gsv) = ρ(g)sf̃(v),

f̃(guv) = ρ(g)uf̃(v),

which implies (iii).

Corollary 6.10. x ∈ G is semisimple ⇐⇒ for any homomorphism φ from G onto aclosed subgroup of some GLn, φ(x) is semisimple. Similarly for unipotent elements.

Jordan Decomposition and F -structures

Let F ⊂ k be a subfield. Assume G is an F -group. Note that if x ∈ G(F ), then xs andxu need not lie in G(F ). Here is an example.

Example 6.11. Assume that char(k) = 2 and F 6= F 2 (i.e., F is non-perfect). Let

G = GL2. Let a ∈ F\F 2 and x =

(0 1a 0

). Then the Jordan decomposition of x in

GL2(k) is

x =

(0 1a 0

)= xsxu =

(√a 0

0√a

)(0 1√

a√a 0

).

But xs, xu 6∈ GL2(F ). Moreover, it is the case that if F is perfect, then the semisimpleand unipotent parts of an element in G(F ) are again in G(F ).

Unipotent Groups

Definition 6.12. A linear algebraic group G is unipotent if all its elements are unipotent.

Example 6.13. The linear group Un =

1 ∗ ∗ · · · ∗0 1 ∗ · · · ∗0 0 1 · · · ∗

0 0 0. . . ∗

0 0 0 · · · 1

is unipotent. Actually

it turns out that this is essentially the only example.

36

Proposition 6.14. Let G be a subgroup of GLn consisting of unipotent matrices. Thenthere exists x ∈ GLn such that xGx−1 ⊂ Un.

Before proving Proposition 6.14, we need the Burnside’s Theorem.

Theorem 6.15. Let E be a finite dimensional vector space over an algebraically closedfield k, R be a subalgebra of End(E). If E is a simple R-module (i.e., the action isirreducible), then R = End(E).

Proof of Proposition 6.14. Put V = kn. If there is some 0 6= W ( V such that G ·W = W , then we have the result by induction on n. Suppose no such W exists, i.e., Gacts irreducibly in V . By Burnside’s Theorem, the elements in G span the vector spaceEnd(V ). Let g ∈ G, then 1 − g is nilpotent, so (1 − g)h is nilpotent for all h ∈ G, henceTr((1 − g)h) = 0 for all h ∈ G. Tr is a linear expression in h, so Tr(h) = Tr(gh) for alln×n matrices h ∈ G. Now choosing h = Eij , we see that this is only possible when g = 1,i.e., G = {1}.

Remark 6.16. By Proposition 6.14, if G is unipotent linear algebraic group and G →GL(V ) is a rational representation of G, then there is a nonzero vector v ∈ V which isfixed by all of G (consider the first basis element after conjugating into Un).

Proposition 6.17 (Kostant-Rosenlicht). Let G be a unipotent linear algebraic group andlet X be an affine G-space. Then all orbits of G in X are closed.

Proof. Let O be an orbit. Without loss of generality we may assume that X = O andhence O is dense in X. Recall that an orbit in open in its closure by Lemma 5.6, so O isopen in O. Let Y = O\O. Then G acts locally finitely on the ideal IX(Y ). Because G isunipotent, we may apply Remark 6.16 to the rational representation G → GL9(IX(Y )).So there is a non-zero function f ∈ IX(Y ) fixed by elements of G, i.e., ρ(g)f = f, ∀g ∈ G.Now for any o ∈ O, (ρ(g)f)(o) = f(o). So f(og) = f(o),∀o ∈ O,∀g ∈ G. Hencef(o) = f(e), ∀o ∈ O, i.e., f is constant on O. Since O is dense in X, f is constant on X.Thus IX(Y ) = k[X], i.e., Y = ∅, and hence O = O.

37

7 Commutative Linear Algebraic Groups (09/30)

Structure of Commutative Algebraic Groups

Theorem 7.1 (Kolchin). Let G be a commutative linear algebraic group. Then(i) The sets Gs and Gu of semisimple and unipotent elements are closed subgroups.(ii) The product map π : Gs ×Gu → G is an isomorphism of algebraic groups.

Proof. (i) We may assume that G is a closed subgroup of some GLn, by Theorem 5.15.Recall that if x, y ∈ End(V ), then xy = yx implies that (xy)s = xsys and (xy)u = xuyu.This implies that both Gs and Gu are subgroups.

Gu is a closed subset for general (not necessarily commutative) linear algebraic groupG because the set of all unipotent matrices in GLn(k) is the zero set of polynomials impliedby (x− 1)n = 0.

To see Gs is closed, recall that without loss of generality we may assume G ⊂ Tn =upper triangular matrices in GLn and Gs ⊂ Dn. This forces Gs = G ∩ Ds which showsthat Gs is closed. (Note that for general G, it is rare that Gs is closed).

(ii) π is an isomorphism of abstract groups by the uniqueness of Jordan decompositionin G. Also, π is a morphism of varieties and the map G → Gs defined by x 7→ xs isa morphism of algebraic varieties because it maps x to some of its matrix entries, so itgives polynomials. Hence π−1 : x 7→ (xs, x

−1s x) is a morphism of varieties. Hence π is an

isomorphism of algebraic groups.

Corollary 7.2. If G is connected, then so are Gs and Gu.

Proof. Gs and Gu are imagies of the connected group G under continuous maps, so theyare connected.

Proposition 7.3. Let G be a connected linear algebraic group of dimension 1. Then(i) G is commutative.(ii) Either G is Gs or Gu.(iii) If G is unipotent and p = char(k) > 0, then the elements of G have order dividing

p.

Proof. (i) Fix g ∈ G and consider the morphism φ : G → G defined by x 7→ xgx−1.Because G is connected (i.e., irreducible topological group), its image φ(G) is also anirreducible topological group, which implies φ(G) is an irreducible closed subset of G. Ifφ(G) is a proper irreducible closed subset of G, it must have dimension less than dimG = 1.So either φ(G) = {g} (i.e., G is commutative) or φ(G) = G.

Let’s assume φ(G) = G. Because φ(G) contains a nonempty open subset of φ(G), wewould have G − φ(G) is finite. Viewing G as a closed subgroup of some GLn, there areonly finitely many possibilities for the characteristic polynomial det(T · 1−x), x ∈ G. ButG is connected, so the characteristic polynomial is constant. Taking x to be identity, itmust be (T − 1)n. This means G is unipotent. Hence G is solvable. Now G′ = (G,G) the

38

commutator subgroup of G is a connected, closed subgroup and can only be {e}. Nowg−1 · φ(G) ⊂ G′, which is a contradiction.

(ii) Because G is connected, both Gs and Gu are irreducible, closed subvarieties of G.If G 6= Gs, then Gs is a proper subvariety so dim(Gs) < dim(G) = 1, i.e., Gs = {e}. ThusG = Gu.

(iii) Assume that G is unipotent and p = char(k) > 0. Let

Gph

= {ph − power of elements of G}.

Then it is easy to check that Gph

is a connected, closed subgroup of G, so it must be G or{e}. Viewing G as an upper triangular matrices in some GLn, Gp

h= {e} if ph ≥ n, which

in characteristic p implies Gp = {e}.

Algebraic Tori

Definition 7.4. Let G be a linear algebraic group. A rational character (or just a char-acter) of G is a homomorphism of algebraic groups χ : G→ Gm. We denote

X∗(G) =abelian group of rational characters with additive notation,

i.e., (χ1 + χ2)(g) = χ1(g)χ2(g).

Note that characters are regular functions on G, so X∗(G) ⊂ k[G]. Also, charactersare linearly independent in k[G] (this is the Dedekind’s Lemma).

Lemma 7.5 (Dedekind’s Lemma). Let G be any group, E be any field. X(G) = Homgroup(G,E∗)is a linearly independent subset of the vector space over E of functions {G→ E}.

Proof. If there is a nontrivial linear independence relation among the elements in X(G),take one of minimal length:

a1χ1 + · · ·+ anχn = 0, 0 6= ai ∈ Z, χidistinct.

For g, h ∈ G,n∑i=1

aiχi(g)χi(h) = 0 = χ1(g)n∑i=1

aiχi(h).

Son∑i=2

ai(χi(g)− χ1(g))χi(h) = 0.

Since χ2 6= χ1, there exists some g ∈ G such that χ2(g)− χ1(g) 6= 0. This contradicts theminimality of the length.

39

Definition 7.6. A cocharacter (or a multiplicative one parameter subgroup) of G is ahomomorphism of algebraic groups Gm → G. We denote

X∗(G) = the set of cocharacters.

Note that cocharacters may not necessarily be abelian. However, if G is commutative,then X∗(G) is an abelian group. Even G is not commutative, we still have an action of Zon X∗(G) by

(n · λ)(a) = λ(a)n.

We write −λ = (−1) · λ.

Definition 7.7. A linear algebraic group G is called diagonalizable if it is isomorphicto a closed subgroup of some Dn. G is called an algebraic torus (or just torus) if it isisomorphic to some Dn.

Example 7.8. G = Dn is a torus, while G = Dn × {±1} is diagonalizable.

Example 7.9. G = Dn =

x1

. . .

xn

xi 6= 0

. Set χi(x) = xi. Then each χi is

a character of Dn and in fact k[Dn] = k[χ1, · · · , χn, χ−11 , · · · , χ−1

n ]. χa11 χa22 · · ·χann , where

(a1, · · · , an) ∈ Zn, are all the characters of Dn, and they form a basis for k[Dn]. Moreover,

X∗(Dn) ∼= Zn

as abelian groups. Also, any cocharacter Gm → Dn is given by

x 7→

xa1

xa2

. . .

xan

where (a1, · · · , an) ∈ Zn. In other words,

X∗(Dn) ∼= Zn

as abelian groups.

Theorem 7.10. The following are equivalent for a linear algebraic group G.(i) G is diagonalizable.(ii) X∗(G) is an abelian group of finite type. X∗(G) is a k-basis for k[G].(iii) Any rational representation of G is a direct sum of one dimensional representa-

tions.

40

Proof. (i) ⇒ (ii). Assume G is diagonalizable. Then G is a closed subgroup of someDn. Hence k[G] is a quotient of k[Dn]. Restriction of characters from Dn to G reducescharacters of G and they span k[G]. By Dedekind’s Lemma (Lemma 7.5), they form a basisand any character of G is a linear combination of these restrictions. Hence X∗(Dn) →X∗(G) is a surjective homomorphism of abelian groups. Recall that X∗(Dn) ∼= Zn. SoX∗(G) is of finite type.

(ii)⇒ (iii). Let φ : G→ GL(V ) be a rational representation of G in a finite dimensionalvector space V . Then (ii) implies that we can define linear maps Aχ : V → V, χ ∈ X∗(G)via φ(x) =

∑χ∈X∗(G) χ(x)Aχ with Aχ = 0 for all but finitely many χ’s. To see this, fix

a basis for V and write φ(x) = [φij(x)]n×n. Then φij ∈ k[G] and by (ii), we can writeφij =

∑χ∈X∗(G) αijχχ. Then φ(x) =

∑χ∈X∗(G)Aχχ(x) where Aχ has the matrix [αijχ]n×n

with respect to the fixed basis. For x, y ∈ G,

φ(xy) =∑

χ∈X∗(G)

χ(xy)Aχ = φ(x)φ(y) =

∑χ∈X∗(G)

χ(x)Aχ

∑χ∈X∗(G)

χ(y)Aχ

.

By Dedekind’s Lemma (Lemma 7.5),

AχAψ =

{0 if χ 6= ψ

Aχ if χ = ψ.

Also,∑

χ∈X∗(G)Aχ = φ(e) = id. Put Vχ = im(Aχ). Then it follows that V is a direct sumof Vχ and x ∈ G acts on Vχ via mapping by χ(x).

(iii) ⇒ (iii). This direction is clear.

Corollary 7.11. If a linear algebraic group G is diagonalizable, then X∗(G) is an abeliangroup of finite type without p-torsion if p = char(k) > 0.

In fact, if G is diagonalizable, the algebra k[G] is isomorphic to the group algebra ofX∗(G).

Group Algebras of Abelian Groups

Let M be an abelian group of finite type. The group algebra of M is

k[M ] := the algebra with basis (em)m∈M with mapping defined by em · en = em+n.

Observe that if M1,M2 are two abelian groups of finite type, then

k[M1 ⊕M2] = k[M1]⊗k k[M2].

Define

∆ : k[m]→ k[M ]⊗k k[M ]

em → em ⊗ em,

41

ι : k[m]→ k[M ]

em → e−m,

and

e : k[m]→ k

em → 1.

Recall that if M is of finite type, then M ∼= Zr ⊕ (direct sum of finite groups). Ifp ·m = 0 for a prime p, then m is called a p-torsion element.

Proposition 7.12. Assume that p = char(k) > 0, and M has no p-torsion.(i) k[M ] is an affine algebra, and there is a diagonalizable group G(M) with k[G(M)] =

k[M ] such that ∆, ι, e are mappings, antipode and the identity element of G(M).(ii) There is a canonical isomorphism M ∼= X∗(G(M)).(iii) If G is diagonalizable, then there is a canonical algebraic group isomorphism

G(X∗(G)) ∼= G.

Example 7.13. Let M = Z⊕Z/2Z = M1⊕M2. Then G(M) = G1×G2 where G1 = G(M1)and G2 = G(M2), and k[M1] ∼= k[T, T−1], k[M2] ∼= k[T ]/(T 12 − 1). By assumption, ifp > 0, p 6 |12, then k[M2] is a reduced algebraic group. Then G(M) ∼= D1 × (finite group).

Example 7.14. G(Zn) ∼= Dn.

Corollary 7.15. Let G be a diagonalizable group.(i) G is a direct product of a torus and a finite abelian group of order prime to p, if

p = char(k) > 0.(ii) G is a torus ⇐⇒ G is connected.(iii) G is a torus ⇐⇒ X∗(G) is a free abelian group.

Proposition 7.16 (Rigidity of Diagonalizable Groups). Let G and H be diagonalizablegroups and let V be a connected affine variety. Assume that ϕ : V ×G→ H is a morphismof varieties such that for each v ∈ V , the map G → H defined by x → ϕ(v, x) is ahomomorphism of algebraic groups. Then ϕ(v, x) is independent of v.

For G an arbitrary linear algebraic group and H a closed subgroup, set

ZG(H) = {g ∈ G : ghg−1 = h,∀h ∈ H}, − centralizer of H in G,

NG(H) = {g ∈ G : gHg−1 ∈ H}, − normalizer of H in G,

The defining conditions can be expressed as polynomial conditions, so these are closedsubgroups of G and ZG(H) / NG(H).

Corollary 7.17. If H is a diagonalizable subgroup of G, then NG(H)0 = ZG(H)0 andNG(H)/ZG(H) is finite.

42

Proof. Let V = NG(H)0. Apply rigidity (Proposition 7.16) to

ϕ : NG(H)0 ×H → H

(x, y) 7→ xyx−1

to conclude that xyx−1 is independent of x, i.e., xyx−1 = y, ∀x ∈ NG(H)0. ThusNG(H)0 ⊂ ZG(H). This proves NG(H)0 = ZG(H)0.

43

8 Tori and Torus Actions (10/07)

Review of Pairings

Let R be a commutative ring with 1 and let M,N be two (left) R-modules. The setHomR(M,N) = {R-linear maps from M to N} is an R-module.

Example 8.1. M is any R-module, N = R, then M∨ = HomR(R,N) is called the dualmodule, or dual space, or R-module of M .

Example 8.2. R∨ = HomR(R,R) ∼= R.

Example 8.3. R = F is a field, M,N are vector spaces over F . This comes from linearalgebra.

Definition 8.4. A pairing between M and N is a bilinear map 〈·, ·〉 : M ×N → R, i.e.,R-linear in each component when the other is fixed.

Example 8.5. A dot product Rn ×Rn → R is a pairing.

Example 8.6. There are two natural pairings,

Mn(R)×Mn(R)→ R

〈A,B〉 = Tr(AB)

and

Mn(R)×Mn(R)→ R

〈A,B〉 = Tr(ABT).

Example 8.7. The map

M ×M∨ → R

〈m,ϕ〉 = ϕ(m)

is called the standard pairing between a module and its dual.

Example 8.8. The map

R[x]×R[x]→ R

〈f, g〉 = f(0)g(0)

is a pairing. Then 〈x, g〉 = 0 for all g ∈ R[x] even though x 6= 0. In fact, 〈f, g〉 = 0 for allg ∈ R[x] if x|f .

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We can use a pairing to think of M and N as part of the dual of the other module.For m ∈ M , n 7→ 〈m,n〉 is a functional on N and for n ∈ N , m 7→ 〈m,n〉 is a functionalon M . However, if the pairing behaves badly, we may have 〈m,n〉 = 0 ∀n with m 6= 0.

For R-modules M and N , HomR(M,N∨), HomR(N,M∨) and

BilR(M,N ;R) = {bilinear maps from M to N}

are all isomorphic as R-modules. The point is that a bilinear map allows us to use M toparametrize a piece of N∨ and similarly for N . However, some pairings may make differentelements of M behave like the same element of N∨. For example, a nonzero element of Mmight pair with every element of N to have the value 0, as behavior we expect if m = 0.

The pairings that allow us to identify M and N with each other’s full dual module arethe “perfect” pairings.

Definition 8.9. A pairing 〈·, ·〉 : M × N → R is called a perfect pairing if the inducedlinear maps M → N∨ and N →M∨ are both isomorphisms.

Note that when R is a field and M , N are finite dimensional vector spaces of the samedimension, then a pairing 〈·, ·〉 : M × N → R is perfect if and only if the induced mapM → N∨ is injective, i.e., 〈m,n〉 = 0 for all n ∈ N implies m = 0. (Then N → M∨

is also automatically an isomorphism.) However, an injective linear map of free moduleswith the same rank need not be an isomorphism, for example, the map Z→ Z defined byx 7→ 2x. So in the case of non-field commutative ring R with M and N free of the samefinite rank, it is not enough to just check that M → N∨ is injective.

Perfect pairing of modules M and N is stronger than just identification of one of themwith the dual of the other. It identifies each module as the dual of the other M ∼= N∨ andN ∼= M∨, both coming from the perfect pairing 〈·, ·〉 : M ×N → R.

Characters and Cocharacters of Tori

Let T be a torus. Denote the character group

X = X∗(T ) = {χ : T → Gm}

and the cocharacter group

Y = X∗(T ) = {λ : Gm → T}.

For χ ∈ X,λ ∈ Y , a ∈ k∗, consider the character

Gm → Gm

a 7→ χ(λ(a)).

Recall X∗(Gm) ∼= Z so χ(λ(a)) = a 〈χ, λ〉 for some 〈χ, λ〉 ∈ Z.

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Lemma 8.10. (i) 〈·, ·〉 : X × Y → Z defines a perfect pairing, i.e., any homomorphismX → Z is of the form χ 7→ 〈χ, λ〉 for some λ ∈ Y and any homomorphism Y → Z is ofthe form λ 7→ 〈χ, λ〉 for some χ ∈ X. In particular, Y is a free Z-module.

(ii) The map a⊗λ 7→ λ(a) defines a canonical isomorphism of abelian groups k∗⊗ZY ∼=T .

Proof. (i) Since T is a torus, it is isomorphic to some Dn. Then X = {χa11 χa22 · · ·χann :

(a1, a2, · · · , an) ∈ Zn} ∼= Zn and Y = {x 7→ diag(xb1xb2 · · ·xbn : (b1, b2, · · · , bn) ∈ Zn} ∼=Zn. So the assertion is clear.

(ii) This follows from the freeness of Y .

Tori and F -structures

Let F ⊂ k be a subfield.

Definition 8.11. An F -torus is an F -group which is also a torus. An F -torus T whichis F -isomorphic to some Dn is called F -split .

Example 8.12. Let k = C, F = R. Then G =

{(a −bb a

)∈ GL2

}is an R-torus which

is not R-split.

Proposition 8.13. (i) An F -torus T is F -split ⇐⇒ all its characters are defined overF . In that case, the characters form a basis of F [T ].

(ii) Any rational representation over F of an F -split torus is a direct sum of one-dimensional representations over F .

Torus Action

Let X,Y, T as before. Let V be an affine T -space. This leads to locally finite representations of T in k[V ] as before.

For χ ∈ X, put

k[V ]χ = {f ∈ k[V ] : s(t) · f = χ(t)f,∀t ∈ T}.

We saw that any rational representation of a diagonalizable group was a direst sum of1-dimensional rational representations of the subspaces k[V ]χ define an X-grading of thealgebra k[V ], i.e., k[V ] = ⊕χ∈Xk[V ]χ and k[V ]χk[V ]ψ ⊂ k[V ]χ+ψ for χ, ψ ∈ X.

Example 8.14. If T = Gm, then X = Z and the grading structure on k[V ] is the usualone (given by degrees of monomials).

For Z a variety and ϕ : Gm → Z a morphism of varieties, write lima→0 ϕ(a) = z if ϕextends to a morphism ϕ̃ : A1 → Z such that ϕ̃(0) = z. Put ϕ′(a) = ϕ(a−1) and definelima→∞ ϕ(a) = lima→0 ϕ

′(a).

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If V is a T -space and λ ∈ T , we write

V (λ) = {v ∈ V : lima→0

λ(a) · v exists}.

ThenV (−λ) = {v ∈ V : lim

a→∞λ(a) · v exists}.

Lemma 8.15. Assume V is affine.(i) V (λ) is a closed subset of V .(ii) V (λ) ∩ V (−λ) is the set of fixed points in Im(λ), i.e.,

V (λ) ∩ V (−λ) = {v ∈ V : λ(k∗) · v = {v}}.

Proof. (i) An element f ∈ V (λ) can be written as f =∑

χ fχ, fχ ∈ k[V ]χ. Then

s(λ(a)) · f =∑χ

a〈χ,λ〉fχ.

So lima→0 λ(a) · v exists ⇐⇒ v annihilates all functions in Vχ with 〈χ, λ〉 < 0. Thisproves (i).

(ii) Now, V (λ) ∩ V (−λ) is the set of v, annihilating all Vχ with 〈χ, λ〉 6= 0. Then

V (λ) ∩ V (−λ) = {v ∈ V : f(λ(a) · v) = f(v), ∀f ∈ k[V ], a ∈ k∗}.

This is just the set of fixed points.

Example 8.16. Let G is a linear algebra group, λ : Gm → G be a cocharacter. Considerthe action of T = Gm on G by a · x = λ(a)xλ(a)−1. Write P (λ) = {x ∈ G : lima→0 a ·x exists}. This is a subgroup of G. By Lemma 8.15 (i), it is closed and by Lemma 8.15(ii), P (λ) ∩ P (−λ) is the centralizer of Im(λ).

Additive Functions

Definition 8.17. An additive function on a linear algebraic group G is a homomorphismof algebraic groups f : G→ Ga. Denote

A = A(G) = the set of additive functions on G,

which is a subspace of the algebra k[G]. Let F ⊂ k be a subfield and G be an F -group,then write

A(F ) = A(G)(F ) = F -vector space of additive functions defined over F .

Note that if p = char(k) > 0, then p-th power of an additive function is again anadditive function. This will allow us to define a ring R over which A is a module.

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